Since \(\operatorname{REGLang}\)  (respectively, \(\operatorname{VPLang}\) ) is closed under union, intersection and complement, the set of languages expressible in \(\operatorname{REG}\)  (respectively, \(\operatorname{VPL}\) ) is exactly \(\operatorname{REGLang}\)  (respectively, \(\operatorname{VPLang}\) ). However, the same is not true for \(\operatorname{DCF}\)  and \(\operatorname{DCFLang}\) , since that class is not closed e.g. under intersection. For \(\operatorname{DCF}\)  the expressible languages are exactly the Boolean closure of the deterministic context-free languages. Moreover, it can be inferred from well-known results on word equations (see e.g. [1]}, [2]}) that the languages expressed by \(\operatorname{WE}\)  are exactly those expressible by a single word equation in the sense of [1]}.
